Conjecture on Brownian tracking horizon for geometric matchings

Let $n$ particles move in $d$ dimensions, sampled at intervals of length $\delta$, and let $\widehat{\pi}_k$ be the estimator at time step $k$. Let $T_{\max}$ denote the maximum tracking time before the estimator incurs a macroscopic number of errors. "Brownian tracking conjecture." Suppose that $d\geq2$ and $\delta=n^{-\xi/d}$ for some $\xi\in[2,4]$. Then

$$T_{\max}\sim\frac{\delta^{1-d/2}}{n}f(n)=n^{\xi/2-\xi/d-1}f(n),$$

for some $1/\operatorname{polylog}(n)\leq f(n)\leq\operatorname{polylog}(n)$. The heuristic is based on Brownian displacements between snapshots and is explicitly not expected to apply in dimension one, where particle collisions dominate the tracking error; the conjectured behavior remains unproved for the stated higher-dimensional regime.